# Mittag-Leffler's theorem

In complex analysis, Mittag-Leffler's theorem concerns the existence of meromorphic functions with prescribed poles. It is sister to the Weierstrass factorization theorem, which asserts existence of holomorphic functions with prescribed zeros. It is named after Gösta Mittag-Leffler.

## Theorem

One possible proof outline is as follows. Notice that if ${\displaystyle E}$ is finite, it suffices to take ${\displaystyle f(z)=\sum _{a\in E}p_{a}(z)}$. If ${\displaystyle E}$ is not finite, consider the finite sum ${\displaystyle S_{F}(z)=\sum _{a\in F}p_{a}(z)}$ where ${\displaystyle F}$ is a finite subset of ${\displaystyle E}$. While the ${\displaystyle S_{F}(z)}$ may not converge as F approaches E, one may subtract well-chosen rational functions with poles outside of D (provided by Runge's theorem) without changing the principal parts of the ${\displaystyle S_{F}(z)}$ and in such a way that convergence is guaranteed.

## Example

Suppose that we desire a meromorphic function with simple poles of residue 1 at all positive integers. With notation as above, letting ${\displaystyle p_{k}=1/(z-k)}$ and ${\displaystyle E={\mathbb {Z} }^{+}}$, Mittag-Leffler's theorem asserts (non-constructively) the existence of a meromorphic function ${\displaystyle f}$ with principal part ${\displaystyle p_{k}(z)}$ at ${\displaystyle z=k}$ for each positive integer ${\displaystyle k}$. This ${\displaystyle f}$ has the desired properties. More constructively we can let ${\displaystyle f(z)=z\sum _{k=1}^{\infty }{\frac {1}{k(z-k)}}}$. This series converges normally on ${\displaystyle \mathbb {C} }$ (as can be shown using the M-test) to a meromorphic function with the desired properties.

Another example is provided by

${\displaystyle {\frac {\pi }{\sin \pi z}}={\frac {1}{z}}+\sum _{k\in \mathbb {Z} ,\,k\neq 0}(-1)^{k}\left({\frac {1}{z-k}}+{\frac {1}{k}}\right).}$

## References

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